Finding eigen vectors of a matrix is the second step in solving a eigenvalue problem. To find eigenvectors, fist we have to find the eigenvalues from the given matrix. Then, we can find the eigenvectors that correspond to each eigenvalue found in the solution of the characteristic equation. It is best to illustrate the procedure with an example.
Example :
Find the eigenvectors of
To find the eigenvectors, first we have to find the eigen values.
To find the eigen values, solve the following equation.
det|A - λI| = 0
where I is the 2 × 2 identity matrix.
The characteristic equation :
(5 - λ)(2 - λ) - 18 = 0
10 - 5λ - 2λ + λ^{2} - 18 = 0
λ^{2 }- 7λ - 8 = 0
(λ - 8)(λ + 1) = 0
λ = 8 or λ = -1
The two eigenvalues of the given matrix :
λ_{1} = 8
λ_{2} = -1
An eigenvector of a 2 × 2 matrix is going to be a column vector with 2 components. If we call these two unknowns x and y, then we can write the vector as
For the first eigenvalue, the eigenvector equation is
Av = λ_{1}v
Specifically, we have the matrix equation,
From the above matrix equation, we have
5x + 2y = 8x -3x + 2y = 0 3x - 2y = 0 ----(1) |
9x + 2y = 8y 9x - 6y = 0 3x - 2y = 0 ----(2) |
The linear equations (1) and (2) above represent the same line. So, the system has infinitely many solutions.
Solve (1) or (2) for y.
y = (³⁄₂)x
If we choose x = 2,
y = (³⁄₂)(2)
y = 3
Then the eigenvector corresponding to λ_{1} = 8 is
For the second eigenvalue, the eigenvector equation is
Av = λ_{2}v
Specifically, we have the matrix equation,
From the above matrix equation, we have
5x + 2y = -x 6x + 2y = 0 3x + y = 0 ----(1) |
9x + 2y = -y 9x + 3y = 0 3x + y = 0 ----(2) |
The linear equations (1) and (2) above represent the same line. So, the system has infinitely many solutions.
Solve (1) or (2) for y.
y = -3x
If we choose x = 1,
y = -3(1)
y = -3
Then the eigenvector corresponding to λ_{1} = -1 is
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